Theorem dmv 5903

Description: The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)

Assertion

Ref	Expression
dmv	⊢ dom V = V

Proof of Theorem dmv

Step	Hyp	Ref	Expression
1		ssv 3963	. 2 ⊢ dom V ⊆ V
2		dmi 5902	. . 3 ⊢ dom I = V
3		ssv 3963	. . . 4 ⊢ I ⊆ V
4		dmss 5883	. . . 4 ⊢ ( I ⊆ V → dom I ⊆ dom V)
5	3, 4	ax-mp 5	. . 3 ⊢ dom I ⊆ dom V
6	2, 5	eqsstrri 3986	. 2 ⊢ V ⊆ dom V
7	1, 6	eqssi 3955	1 ⊢ dom V = V

Syntax hints:   = wceq 1563  Vcvv 3457   ⊆ wss 3907   I cid 5546  dom cdm 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-dm 5662

This theorem is referenced by:  nowisdomv  30734